I was trying to do a one handed handstans type thing, and my elbow twisted and I fell backwards. There was a slight ripping sound. When at rest, my elbow does not hurt. I can put weight on it in certain positions, and do a push up pain free. However when I move my arm in certain ways, the muscle under the elbow and the elbow itself hurt. It also hurts somewhat to straighten my arm. The arm and elbow itself look normal. Is this a bone, muscle, or ligament/tendon related problem? What should I do?

You should go to the doctor.

No one can diagnose this without an examination and possibly some x-rays.